A078987 Chebyshev U(n,x) polynomial evaluated at x=19.

1, 38, 1443, 54796, 2080805, 79015794, 3000519367, 113940720152, 4326746846409, 164302439443390, 6239165952002411, 236924003736648228, 8996872976040630253, 341644249085807301386, 12973484592284636822415, 492650770257730391950384, 18707755785201470257292177

Offset: 0

Author

Wolfdieter Lang, Jan 10 2003

Keywords

• nonn
• easy

Comments

A078986(n+1)^2 - 10*(6*a(n))^2 = +1, n>=0 (Pell equation +1, see A033313 and A033317).

a(n) equals the number of 01-avoiding words of length n on alphabet {0,1,...,37}. - Milan Janjic, Jan 26 2015

Links

• Colin Barker, Table of n, a(n) for n = 0..632
• Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
• R. Flórez, R. A. Higuita, and A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
• Tanya Khovanova, Recursive Sequences
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (38,-1).

Crossrefs

Cf. A097314, A097315.

Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), A007655 (m=7), A077412 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), this sequence (m=19), A097316 (m=33).

Programs

GAP

m:=19;; a:=[1,2*m];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 22 2019

Magma

m:=19; I:=[1, 2*m]; [n le 2 select I[n] else 2*m*Self(n-1) -Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 22 2019

Maple

seq( simplify(ChebyshevU(n, 19)), n=0..20); # G. C. Greubel, Dec 22 2019

Mathematica

lst={};Do[AppendTo[lst, GegenbauerC[n, 1, 19]], {n, 0, 8^2}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)
ChebyshevU[Range[21] -1, 19] (* G. C. Greubel, Dec 22 2019 *)

PARI

a(n)=subst(polchebyshev(n,2),x,19) \\ Charles R Greathouse IV, Feb 10 2012

PARI

Vec(1/(1-38*x+x^2) + O(x^50)) \\ Colin Barker, Jun 15 2015

Sage

[lucas_number1(n,38,1) for n in range(1, 16)] # Zerinvary Lajos, Nov 07 2009

Sage

[chebyshev_U(n,19) for n in (0..20)] # G. C. Greubel, Dec 22 2019

Formula

a(n) = 38*a(n-1) - a(n-2), n>=1, a(-1)=0, a(0)=1.

a(n) = S(n, 38) with S(n, x) = U(n, x/2), Chebyshev's polynomials of the second kind. See A049310.

G.f.: 1/(1-38*x+x^2).

a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*38^(n-2*k).

a(n) = ((19+6*sqrt(10))^(n+1) - (19-6*sqrt(10))^(n+1))/(12*sqrt(10)).

a(n) = Sum_{k=0..n} A101950(n,k)*37^k. - Philippe Deléham, Feb 10 2012

Product_{n>=0} (1 + 1/a(n)) = 1/3*(3 + sqrt(10)). - Peter Bala, Dec 23 2012

Product_{n>=1} (1 - 1/a(n)) = 3/19*(3 + sqrt(10)). - Peter Bala, Dec 23 2012

From Andrea Pinos, Jan 02 2023: (Start)

a(n) = (A097314(n+1) - A097315(n+1))/2.

a(n) = (A097314(n) + A097315(n))/2. (End)